Hall–Littlewood Polynomials, Boundaries, and <i>p</i>-Adic Random Matrices

Peski, Roger Van

doi:10.1093/imrn/rnac143

Cited by 9 publications

(4 citation statements)

References 80 publications

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“…In recent years the moment method was successfully used in many situations [32,19,14,2,35,3,4,12,33] see also the survey of Wood [40]. Note that some of these results can be also extended to symmetric matrices, where in the limit we get some modified version of the Cohen-Lenstra distribution [5,6].…”

Section: Background and Discussionmentioning

confidence: 99%

The local weak limit of 𝑘-dimensional hypertrees

Mészáros¹

2022

Preprint

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Section: Background and Discussionmentioning

confidence: 99%

The local weak limit of 𝑘-dimensional hypertrees

Mészáros¹

2022

Preprint

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“…When the principal specialization is infinite, nice formulas for the principally specialized skew Hall-Littlewood polynomials were shown in [54]. The phrasing below follows [77,Theorem 3.3]. Proposition 3.11.…”

Section: Symmetric Functionsmentioning

confidence: 99%

“…Hence the singular numbers of a product A τ • • • A 1 , with A i ∈ Mat N (Z p ) additive Haar, will become larger as τ increases, looking like Figure 3 instead of Figure 2. For fixed τ , an exact distribution was computed in [76,Corollary 3.4], further simplified in [77,Theorem 1.4], and shown to be universal in the N → ∞ limit for matrices with generic iid entries in [68]. For fixed N and τ → ∞, the singular numbers were shown in [76] to have independent Gaussian fluctuations asymptotically.…”

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confidence: 99%

Limits and fluctuations of p-adic random matrix products

Peski¹

2021

Sel. Math. New Ser.

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We study the distribution of singular numbers of products of certain classes of padic random matrices, as both the matrix size and number of products go to ∞ simultaneously. In this limit, we prove convergence of the local statistics to a new random point configuration on Z, defined explicitly in terms of certain intricate mixed q-series/exponential sums. This object may be viewed as a nontrivial p-adic analogue of the interpolating distributions of Akemann-Burda-Kieburg [6], which generalize the sine and Airy kernels and govern limits of complex matrix products. Our proof uses new Macdonald process computations and holds for matrices with iid additive Haar entries, corners of Haar matrices from GLN (Zp), and the p-adic analogue of Dyson Brownian motion studied in [80]. Contents 1. Introduction 1 2. p-adic matrix background 3. Symmetric function background 4. The limit of the Plancherel/principal Hall-Littlewood measure 5. Residue expansions 6. Tightness and the limiting random variable 7. An indeterminate moment problem 8. Examples of residue formula for L k,t,χ 9. The case of pure α specializations 10. From processes of infinitely many particles to finite matrix bulk limits Appendix A. Parallels with complex matrix products Appendix B. A Hall-Littlewood proof of [80, Theorem 1.2] References

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“…In the case of the Gelfand-Tsetlin graph it is equivalent to the classification of extreme characters of the infinite-dimensional unitary group U(∞) and also the classification of totally non-negative Toeplitz matrices, see [31]. The fact that an explicit classification sometimes exists is remarkable and has been achieved only for a handful of models, see for example [31,127,126,115,114,125,101,69,70,11]. In the case of GT + , recall a x ≡ 1, all such measures are given by (22), see [31].…”

Section: Extremal Measures For the Inhomogeneous Gelfand-tsetlin Graphmentioning

confidence: 99%

Determinantal Structures in Space-Inhomogeneous Dynamics on Interlacing Arrays

Assiotis

2020

Ann. Henri Poincaré

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We introduce a space inhomogeneous generalization of the dynamics on interlacing arrays considered by Borodin and Ferrari in [7]. We show that for a certain class of initial conditions the point process associated to the dynamics has determinantal correlation functions and we calculate explicitly, in the form of a double contour integral, the correlation kernel for one of the most classical initial conditions, the densely packed. En route to proving this we obtain some results of independent interest on non-intersecting general pure-birth chains, that generalize the Charlier process, the discrete analogue of Dyson's Brownian motion. Finally, these dynamics provide a coupling between the inhomogeneous versions of the TAZRP and PushTASEP particle systems which appear as projections on the left and right edges of the array respectively⋆ ⋆ + 1 ⋆ + 2 ⋆ + 3 ⋆ + 4 ⋆ + 5 ⋆ + 6 Figure 1: A configuration of particles in GT 4 . If the clock of the particle labelled X 3,1 4 rings, which happens at rate λ(⋆ + 1), then the move is blocked since interlacing with X 2,1 4 would be violated. On the other hand, if the clock of the particle X 2,2 4 rings, which happens at rate λ(⋆ + 3), then it jumps to the right by one and instantaneously pushes both X 3,3 4 and X 4,4 4 to the right by one as well, for otherwise the interlacing would break.The right edge particle system is called inhomogeneous PushTASEP, see [7], [6] and also [16] for a q-deformation of the homogeneous case. The fact that this particle system has an underlying determinantal structure is new as well, but is also obtained in the independent work of Petrov [46] that uses different methods.

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Hall–Littlewood Polynomials, Boundaries, and p-Adic Random Matrices

Cited by 9 publications

References 80 publications

The local weak limit of 𝑘-dimensional hypertrees

The local weak limit of 𝑘-dimensional hypertrees

Limits and fluctuations of p-adic random matrix products

Determinantal Structures in Space-Inhomogeneous Dynamics on Interlacing Arrays

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